ICSE QUESTION PAPER Class X Maths (2016) Solution
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1 ICSE QUESTION PAPER Class X Maths (016) Solution SECTION A 1. (a) Let f(x) x x kx 5 Using remainder theorem, f() 7 () () k() 5 7 (8) (4) k() k 5 7 k k 6 k 1 (b) A = 9A + MI A 9A mi... (1) A AA Substitute A in (1) A 9A mi m m m 14 0 m m m 14
2 (c) Sum of all observations Mean = Total number of observations x = x 68 = 9 61= x x = x = 115 Data in ascending order 6, 45, 5, 60, 69, 70, 81, 94, 115 n +1 Since the number of observations is odd, the median is the th 9+1 th Median observation = 5 observation Hence, the median is 69. k. (a) (i) Slope of PQ = 1 k 6 1 k k 5 k 5 ( k) k 5 6 k k 11 (ii) Substituting k in P and Q, we get P(6, k) = P 6, 11 and Q(1 k, ) = Q(4, ) Midpoint of PQ =, =, = 0, 4 (b) cosec 57 tan cos44 cosec 46 cos45 tan 60 th observation cosec (90 ) tan cos44cosec(90 44 ) cos45 tan 60 sec tan cos44 sec44 cos45 tan cos45 tan
3 (c) Let the number of cones be n. Let the radius of sphere be r = 6 cm Radius of a cone be r = cm πr s = n πrc h πr s = n πrc h s c c And height of the cone be h = cm Volume of sphere = n(volume of a metallic cone) r = n r h 4(6) n = () () 4 16 n = 4 n = 7 Hence, the number of cones is 7. x1. (a) (x 7) 15 7x x 1 (x 7) 15 7x and 15 7x x x and 45 1x x 1 x 7x 15 1 and 45 1 x 1x 4x 6 and 44 x x and x x 1.5 and x The solution set is x : xr, 1.5 x. s
4 (b) (i) AD is parallel to BC, i.e., OD is parallel to BC and BD is transversal. (c) ODB = CBD =... Alternate angles In OBD, OD = OB ODB = OBD =...(Radii of the same circle) (ii) AD is parallel to BC, i.e., AO is parallel to BC and OB is transversal. AOB = OBC... Alternate angles OBC = OBD + DBC OBC = + OBC = 64 AOB = 64 (iii) In OAB, OA = OB....(Radii of the same circle) OAB = OBA = x (say) OAB + OBA + AOB = 180 x + x + 64 = 180 x = x = 116 x = 58 OAB = 58 i.e., DAB = 58 DAB = BED = 58...( Angles inscribed in the same arc are equal. ) a +b 18 = 5a +b 9 9(a + b) = 18(5a + b) 87a + 58b = 90a + 54b 58b 54b = 90a 87a 4b = a a 4 = b
5 4. (a) Total number of outcomes = 0 (i) The perfect squares lying between 11 and 40 are 16, 5 and 6. So the number of possible outcomes Probability that the number on the card drawn is a perfect square Number of possible outcomes 1 = = = Total number of outcomes 0 10 (ii) The numbers from 11 to 40 that are divisible by 7 are 14, 1, 8 and 5. So the number of possible outcomes 4 Probability that the number on the card drawn is divisible by 7 Number of possible outcomes 4 = Total number of outcomes 0 15 (b) (i) A = (4, 4) AND B = (, 0) (ii) The figure is an arrow head. (iii) The y-axis is the line of symmetry of figure OABCB A.
6 (c) (i) P = Rs. 5000, T = 1 year, A =Rs. 55 I = A P I = I = 5 So, the interest at the end of first year is Rs. 5. PRT I = 100 I 100 R = P T R = R = = 6.5% 5000 So,the rate of interest at the end of the first year i s 6.5%. (ii) The amount at the end of the first year will be the principal for the second year. P = Rs. 55, T = 1 year, R = 6.5% PRT I = I = 100 I = A = P + I A = A = A Rs So,the amount at the end of the second year is Rs
7 5. (a) x (x + ) = 0 x x 9 = 0 SECTION B (40 Marks) Attempt any four questions from this section Comparing with ax bx c, we get a = 1, b =, c = 9 x= b ± b 4ac a x= (1) ( )± ( ) 4(1)( 9) ± 9+6 x= ± 45 x= ± 9 5 x= ± 5 x= x = or x = x = or x = x = or x = x = or x = x = 4.85 or x = 1.85
8 (b) Since the interest is earned on the minimum balance between 10 th day and the last day of the month as per entries, we have Minimum balance for April = Rs. 800 Minimum balance for May = Rs Minimum balance for June = Rs Minimum balance for July = Rs Minimum balance for August = Rs. 900 Minimum balance for September = Rs. 0 Total balance = Rs Total amount qualifying for interest = Rs Principle for 1 month year = Rs , 1 Rate of interest = 4.5% per annum Interest = Rs. 1 = Rs Amount = Balance in the account in last month + Interest = = Rs Thus, Mrs. Ravi receives Rs on closing the account. (c) Given P = Rs. 1500, I = , R = 10% A = P + I A = Rs Rs = Rs R A = P = = = 1.1 n n n n n n
9 6. (a) Steps of construction: 1. Draw AF measuring 5 cm using a ruler.. With A as the centre and radius equal to AF, draw an arc above AF.. With F as the centre, and same radius cut the previous arc at Z 4. With Z as the centre, and same radius draw a circle passing through A and F. 5. With A as the centre and same radius, draw an arc to cut the circle above AF at B. 6. With B as the centre and same radius, draw ar arc to cut the circle at C. 7. Repeat this process to get remaining vertices of the hexagon at D and E. 8. Join consecutive arcs on the circle to form the hexagon. 9. Draw the perpendicular bisectors of AF, FE and DE. 10. Extend the bisectors of AF, FE and DE to meet CD, BC and AB at X, L and O respectively. 11. Join AD, CF and EB. 1. These are the 6 lines of symmetry of the regular hexagon. 5 cm
10 (b) (i) Since PQRS is a cyclic quadrilateral, QRT SPT...(1)(exterior angle is equal to interior opposite angle) In TPS and TRQ, PTS = RTQ...(common angle) QRT = SPT...(from 1) TPS TRQ...(AA similarity criterion) (ii) Since TPS SP TP i.e., = QR TR SP 18 = SP = 6 SP = 1 cm TRQ, implies that corresponding sides are proportional (iii) Since TPS TRQ, Ar( TPS) SP = Ar( TRQ) RQ 7 1 = Ar( TRQ) Ar( TRQ)= 1 1 Ar( TRQ)= cm Now, Ar( PQRS) = Ar( TPS) Ar( TRQ)= 7 4 cm (c) o o A= 4sin0 cos0 and B= 4 o o 5 cos0 4sin0 (i) Let the order of matrix X mn Order of matrix A Order of matrix B 1 Now, AX B Order of matrix X = m n =
11 x (ii) Let the matrix X = y AX = B o o 4sin0 cos0 x 4 = o o cos0 4sin0 y x 4 = 1 y x 4 = 1 y 5 x + y 4 = x +y 5 x + y = 4...(1) x +y = () Multiplying (1) by, we get 4x +y = 8...() Subtracting () from (), we have x x1 Substitute x in (1), we get 1 y 4 y x 1 Hence, the matrix X = = y
12 7. (a) A is the aeroplane, D and C are the ships sailing towards A. Ships are sailing towards the aeroplane in the same direction. In the figure, height AB=1500 m To find: Distance between the ships, that is CD. Solution: In the right-angled ABC, AB tan 45 = BC = BC BC= 1500 m In the right-angled ABD, AB tan0 = BD BD BD = 1500 m BD = 1500(1.7) = 598 m Distance between the ships = CD = BD BC = = 1098 m
13 (b) Scores f c.f n = 160 th th n 160 th (i) Median= term = term =80 term Through mark 80 on y-axis, draw a horizontal line which meets theogive drawn at point Q. Through Q, draw a vertical line which meets thex-axis at the mark of 4. Median = 4
14 (ii) Since the number of terms = 160 th 160 th Lower quartile (Q 1 ) term 40 term 8 4 th 160 th Upper quartile (Q ) term 10 term 60 4 Inter-quartile range Q Q (a) (iii) Since 85% scores = 85% of 100 = 85 Through mark for 85 on x-axis, draw a vertical line which meets the ogive drawn at point B. Through the point B, draw a horizontal line which meets the y-axis at the mark of 150. Number of shooters who obtained more than 85% score= =10 x y z Let = = = k a b c x ak, y bk, z ck x y z L.H.S. a b c ak bk ck a b c a k b k c k a b c k k k k xyz R.H.S. abc (ak)(bk)(ck) abc k L.H.S. L.H.S. = R.H.S. x y z xyz + + = a b c abc
15 (b) Steps for construction: (i) Draw AB = 5 cm using a ruler. (ii) With A as the centre cut an arc of cm on AB to obtain C. (iii) (iv) (v) (vi) With A as the centre and radius.5 cm, draw an arc above AB. With same radius, and C as the centre draw an arc to cut the previous arc and mark the intersection as O. With O as the centre and radius.5 cm, draw a circle so that points A and C lie on the circle formed. Join OB. (vii) Draw the perpendicular bisector of OB to obtain the mid-point of OB, M. (viii) With the M as the centre and radius equal to OM, draw a circle to cut the previous circle at points P and Q. (ix) Join PB and QB. PB and QB are the required tangents to the given circle from exterior point B. QB = PB = cm That is, length of the tangents is cm.
16 (c) (i) Since, A lies on the x-axis, let the coordinates of A be (x, 0). Since B lies on the y - axis, let the coordinates of B be (0, y). Let m = 1 and n =. Using section formula, 1(0) + (x) 1y + (0) Coordinates of P =, x y (4, 1) =, x y = 4 and = 1 x = 6 and y = So, the coordinates of A are (6, 0) and that of B are (0, ). 0 1 (ii) Slope of AB = = = Slope of line perpendicular to AB = m = P =(4, 1) Required equation is y y = m(x x ) 1 1 y ( 1)= (x 4) y +1 = x + 8 x + y = 7
17 9. (a) Marked price of the article = Rs. 6,000. A dealer buys an article at a discount of 0% from the wholesaler. Price of the article which the dealer paid to the wholesaler % of Rs Sales tax paid by dealer 6% of Rs Amount of the article inclusive of sales tax at which the dealer bought it = Rs.400 Rs.5 = Rs.445 Dealer sells the article at a discount of 10% to the shopkeeper. Price of the article which the shopkeeper paid to the dealer % of Rs Sales tax paid by shopkeeper 6% of Rs Amount of the article inclusive of sales tax at which the shopkeeper bought it Rs.5400 Rs.4 = Rs.574 The value added by dealer Rs.5400 Rs.400 Rs.100 Amount of VAT paid by dealer = 6% of Rs. 7 Price paid by shopkeeper including tax is Rs 574. VAT paid by dealer is Rs.7.
18 (b) (i) Area of the shaded part Area of the circle area of the semicircle () (.5) [6.5 ] [8.5] 7 6 cm product of the diagonals ACBD 18 (ii) Area of kite 48 cm Area of the unshaded part Area of the kite Area of the shaded part 1 (c) (i) Scale factor k cm Length of the model of the ship = k Length of the ship 1 = Length of the ship 00 Length of the ship = 600 m (ii) Area of the deck of the model k Area of the deck of the ship 1 Area of the deck of the model 180, , m ( iii) Volume of the model = k Volume of the ship Volume of the ship 00 Volume of the ship = m
19 10. (a) (i) I=Rs. 100, n 1 4months, r 6% n(n 1) r I P P P 100 P P Rs. 800 So the monthly instalment is Rs (ii) Total sum deposited = P n = Rs Rs Amount of maturity Total sum deposited Interest on it = Rs Rs. 100 Rs (b) (i) Class interval Frequency (ii) Class interval Frequency Mean value fx (f) (x) f = 5 f = 695 fx 695 Mean = = = 7.8 f 5 (iii) Here the maximum frequency is 8 which is corresponding to class 0 0. Hence, the modal class is 0 0.
20 40 (c) Time taken by bus to cover total distance with speed x km/h x 40 Time taken by bus to cover total distance with speed (x 10) km/h x 10 According to the given condition, = x 10 x x 10 x x 10 x 10 x x 10 1 x(x 10) x 10x 10 x 10x 100 x 10x (x 40)(x 0) 0 x 40 0 or x 0 0 x 40 or x 0 Since, the speed cannot be negative, the uniform speed is 40 km/h. 11. (a) cosa L.H.S. = + tana 1 + sina cos A(1 sina) sina (1 sina)(1 sina) cosa cosa sinacosa sina 1 sin A cosa cosa sinacosa sina = cos A cosa 1 sina sina = cosa cosa cosa 1 = cosa = seca = R.H.S
21 (b) (i) Steps of construction: 1. Draw BC = 6.5 cm using a ruler.. With B as the centre and radius equal to approximately half of BC, draw an arc that cuts the segment BC at Q.. With Q as the centre, and same radius, cut the previous arc at P. 4. Join BP and extend it. 5. With B as the centre and radius 5 cm, draw an arc that cuts the arm PB to obtain point A. 6. Join AC to obtain ΔABC. (ii) Steps for construction : 1. With A as the centre and radius.5 cm, draw a circle.. The circumference of a circle is the required locus.
22 (iii) Steps for construction : 1. With C as the centre and with radius of a length less than CA or BC, draw an arc to cut the line segments AC and BC at D and E respectively.. With the same radius or a suitable radius and with D as the centre, draw an arc of a circle.. With the same radius and with E as the centre draw an arc such that the two arcs intersect at H. 4. Join C and H. 5. CH is the bisector of ÐACB and is the required locus. (iv) Steps for construction : 1. We known that the points at a distance of.5 cm from A will surely lie on the circle with centre A.. Also, the points on the angle bisector CH are the points equidistant from AC and BC.. Mark X and Y which are at the intersection of the circle and the angle bisector CH. 4. Measure XY. XY = 5 cm (c) (i) Total dividend = Rs.,475 1 Dividend on each share 1% of Rs. 5= 5 Rs. 100 Total dividend 475 Number of shares bought 85 Dividend on 1 share Total Investment 6400 (ii) Market value of each share = = = Rs. Number of shares bought 85
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